The thesis deals with so-called evolutionary equations, a class of abstract linear operator equations, which cover a huge class of partial differential equation with and without memory. We provide a unified Hilbert space framework for the well-posedness of such equations. Moreover, we inspect the exponential stability of those problems and construct spaces of admissible inital values and pre-histories, on which a strongly continuous semigroup could be associated with the given problem. The theoretical results are illustrated by several examples.

Cell migration, the active translocation of cells is involved in various biological processes, e.g. development of tissues and organs, tumor invasion and wound healing. Cell migration behavior can be divided into two distinct classes: single cell migration and collective cell migration. Single cell migration describes the migration of cells without interaction with other cells in their environment. Collective cell migration is the joint, active movement of multiple cells, e.g. in the form of strands, cohorts or sheets which emerge as the result of individual cell-cell interactions. Collective cell migration can be observed during branching morphogenesis, vascular sprouting and embryogenesis. Experimental studies of single cell migration have been extensive. Collective cell migration is less well investigated due to more difficult experimental conditions than for single cell migration. Especially, experimentally identifying the impact of individual differences in cell phenotypes on individual cell migration behavior inside cell populations is challenging because the tracking of individual cell trajectories is required. In this thesis, a novel mathematical modeling approach, individual-based lattice-gas cellular automata (IB-LGCA), that allows to investigate the migratory behavior of individual cells inside migrating cell populations by enabling the tracking of individual cells is introduced. Additionally, stochastic differential equation (SDE) approximations of individual cell trajectories for IB-LGCA models are constructed. Such SDE approximations allow the analytical description of the trajectories of individual cells during single cell migration. For a complete analytical description of the trajectories of individual cell during collective cell migration the aforementioned SDE approximations alone are not sufficient. Analytical approximations of the time development of selected observables for the cell population have to be added. What observables have to be considered depends on the specific cell migration mechanisms that is to be modeled. Here, partial integro-differential equations (PIDE) that approximate the time evolution of the expected cell density distribution in IB-LGCA are constructed and coupled to SDE approximations of individual cell trajectories. Such coupled PIDE and SDE approximations provide an analytical description of the trajectories of individual cells in IB-LGCA with density-dependent cell-cell interactions. Finally, an IB-LGCA model and corresponding analytical approximations were applied to investigate the impact of changes in cell-cell and cell-ECM forces on the migration behavior of an individual, labeled cell inside a population of epithelial cells. Specifically, individual cell migration during the epithelial-mesenchymal transition (EMT) was considered. EMT is a change from epithelial to mesenchymal cell phenotype which is characterized by cells breaking adhesive bonds with surrounding epithelial cells and initiating individual migration along the extracellular matrix (ECM). During the EMT, a transition from collective to single cell migration occurs. EMT plays an important role during cancer progression, where it is believed to be linked to metastasis development. In the IB-LGCA model epithelial cells are characterized by balanced cell-cell and cell-ECM forces. The IB-LGCA model predicts that the balance between cell-cell and cell-ECM forces can be disturbed to some degree without being accompanied by a change in individual cell migration behavior. Only after the cell force balance has been strongly interrupted mesenchymal migration behavior is possible. The force threshold which separates epithelial and mesenchymal migration behavior in the IB-LGCA has been identified from the corresponding analytical approximation. The IB-LGCA model allows to obtain quantitative predictions about the role of cell forces during EMT which in the context of mathematical modeling of EMT is a novel approach.

Cell migration, the active translocation of cells is involved in various biological processes, e.g. development of tissues and organs, tumor invasion and wound healing. Cell migration behavior can be divided into two distinct classes: single cell migration and collective cell migration. Single cell migration describes the migration of cells without interaction with other cells in their environment. Collective cell migration is the joint, active movement of multiple cells, e.g. in the form of strands, cohorts or sheets which emerge as the result of individual cell-cell interactions. Collective cell migration can be observed during branching morphogenesis, vascular sprouting and embryogenesis. Experimental studies of single cell migration have been extensive. Collective cell migration is less well investigated due to more difficult experimental conditions than for single cell migration. Especially, experimentally identifying the impact of individual differences in cell phenotypes on individual cell migration behavior inside cell populations is challenging because the tracking of individual cell trajectories is required. In this thesis, a novel mathematical modeling approach, individual-based lattice-gas cellular automata (IB-LGCA), that allows to investigate the migratory behavior of individual cells inside migrating cell populations by enabling the tracking of individual cells is introduced. Additionally, stochastic differential equation (SDE) approximations of individual cell trajectories for IB-LGCA models are constructed. Such SDE approximations allow the analytical description of the trajectories of individual cells during single cell migration. For a complete analytical description of the trajectories of individual cell during collective cell migration the aforementioned SDE approximations alone are not sufficient. Analytical approximations of the time development of selected observables for the cell population have to be added. What observables have to be considered depends on the specific cell migration mechanisms that is to be modeled. Here, partial integro-differential equations (PIDE) that approximate the time evolution of the expected cell density distribution in IB-LGCA are constructed and coupled to SDE approximations of individual cell trajectories. Such coupled PIDE and SDE approximations provide an analytical description of the trajectories of individual cells in IB-LGCA with density-dependent cell-cell interactions. Finally, an IB-LGCA model and corresponding analytical approximations were applied to investigate the impact of changes in cell-cell and cell-ECM forces on the migration behavior of an individual, labeled cell inside a population of epithelial cells. Specifically, individual cell migration during the epithelial-mesenchymal transition (EMT) was considered. EMT is a change from epithelial to mesenchymal cell phenotype which is characterized by cells breaking adhesive bonds with surrounding epithelial cells and initiating individual migration along the extracellular matrix (ECM). During the EMT, a transition from collective to single cell migration occurs. EMT plays an important role during cancer progression, where it is believed to be linked to metastasis development. In the IB-LGCA model epithelial cells are characterized by balanced cell-cell and cell-ECM forces. The IB-LGCA model predicts that the balance between cell-cell and cell-ECM forces can be disturbed to some degree without being accompanied by a change in individual cell migration behavior. Only after the cell force balance has been strongly interrupted mesenchymal migration behavior is possible. The force threshold which separates epithelial and mesenchymal migration behavior in the IB-LGCA has been identified from the corresponding analytical approximation. The IB-LGCA model allows to obtain quantitative predictions about the role of cell forces during EMT which in the context of mathematical modeling of EMT is a novel approach.

The thesis deals with so-called evolutionary equations, a class of abstract linear operator equations, which cover a huge class of partial differential equation with and without memory. We provide a unified Hilbert space framework for the well-posedness of such equations. Moreover, we inspect the exponential stability of those problems and construct spaces of admissible inital values and pre-histories, on which a strongly continuous semigroup could be associated with the given problem. The theoretical results are illustrated by several examples.